Inward and Outward Integral Equations and the KKR Method for Photons

TL;DR

This paper develops inward and outward integral equations for electromagnetic waves, derives a photonic KKR method from a variational principle, and discusses its numerical implementation and matching rules for field derivatives.

Contribution

It introduces a variational derivation of the photonic KKR method and clarifies the role of inward equations for electromagnetic wave analysis.

Findings

Inward and outward integral equations are derived for electromagnetic waves.

The variational principle leads to a consistent photonic KKR method.

Surface integral formulation is limited to constant permeabilities.

Abstract

In the case of electromagnetic waves it is necessary to distinguish between inward and outward on-shell integral equations. Both kinds of equation are derived. A correct implementation of the photonic KKR method then requires the inward equations and it follows directly from them. A derivation of the KKR method from a variational principle is also outlined. Rather surprisingly, the variational KKR method cannot be entirely written in terms of surface integrals unless permeabilities are piecewise constant. Both kinds of photonic KKR method use the standard structure constants of the electronic KKR method and hence allow for a direct numerical application. As a by-product, matching rules are obtained for derivatives of fields on different sides of the discontinuity of permeabilities. Key words: The Maxwell equations, photonic band gap calculations